Collapsing Pressure of Thin-walled Cylinders

7/28/2019 Collapsing Pressure of Thin-walled Cylinders

1/86

HI LL IN I SUNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

PRODUCTION NOTEUniversity of Illinois at

Urbana-Champaign LibraryLarge-scale Digitization Project, 2007.


2/86


3/86

UNIVERSITY OF ILLINOISBULLETINVol XXXIX November 11, 1941 No. 12

ENGINEERING EXPERIMENT STATIONBULLETIN SERIES No. 329

A STUDY OF THE COLLAPSING PRESSUREOF THIN-WALLED CYLINDERS

ROLLAND. STURMROLLAND G. STURM

Price: $1.50PUBLISHED BY THE UNIVERSITY OF ILLINOISURBANA

[Issued weekly. Entered as second-cleas matter December 11, 1912, at the post offile at Urbans, Illinois,under the Act of August 24, 1912. Acceptance for mailin at the special rate of postae provided for insection 1108, Act of October 3, 1917, authorised July 31, 19181


4/86


5/86

UNIVERSITY OF ILLINOISENGINEERING EXPERIMENT STATION

BULLETIN SERIES NO. 329

A STUDY OF THE COLLAPSING PRESSUREOF THIN-WALLED CYLINDERS

BY

ROLLAND GEORGE STURMRESEARCH ENGINEER PHYSICIST ALUMINUM

COMPANY OF AMERICA(Formerly Graduate Student of University of Illinois)

PUBLISHED BY THE UNIVERSITY OF ILLINOIS


6/86


7/86

CONTENTSPAGE

I. INTRODUCTION . . . . . . . . .1. R6sum6 of Previous Work . . . .2. Purpose and Scope of Investigation .3. Acknowledgment . . . . . . .

II. THEORETICAL ANALYSIS . . . . . . .4. Basic Arguments and Assumptions5. System of Coordinates6. Notation . . . . . .7. Deformed Element of Shell . . . .8. Internal Forces and Couples . . . .9. Internal Forces and Couples in Terms of

Displacement . . . . . . .10. Equations of Equilibrium . . . . .11. General Differential Equations .

. . . 7

788

99101013. . . 14

15. . . 16. . . 17

III. SOLUTION OF BUCKLING EQUATIONS FOR ROUNDCYLINDERS, WITHOUT STIFFENERS . . . . . . 19

12. Uniform Pressure Applied to Sides Only . . . 19(a) Edges of Shell at Ends Simply Supported 20(b) Edges of Shell at Ends Fixed . . . . . 25(c) Edges of Shell at Ends Restrained . . . 30

13. Pressure Applied to Sides and Ends . . . . . 30(a) Edges of Shell at Ends Simply Supported . 31(b) Additional End Load . . . . . . . 32(c) Edges of Shell at Ends Fixed . . . . . 33(d) Edges of Shell at Ends Restrained . . . 33

14. Pressure on Ends Only . . . . . . . . 34

IV. ROUND CYLINDERS STIFFENED WITH RINGS .15. Shell Round at Each Ring . . . .

3636


8/86

4 CONTENTS (CONCLUDED)PAGE

V. CYLINDERS SLIGHTLY OUT-OF-ROUND . . . . . . 3716. Out-of-Round Unstiffened Cylinders . . . . 3717.. Out-of-Round Cylinders Stiffened With Rings . 40

VI. EXTENSION TO PLASTIC ACTION . . . . . . . . 4118. Cylinders in Which Stresses Are Beyond

Elastic Range . . . . . . . . . . 41

VII. EXPERIMENTAL WORK . . . . . . . . . . 4419. Effects Considered . . . . . . . . . . 44

A. Tests at the University of Illinois . . . . 4520. Object of Tests . . . . . . . . . . . 4521. Specimens . . . . . . . . . . . . 4522. Apparatus . . . . . . . . . . . . 4723. Discussion of Tests and Results . . . . . . 48

B. Tests at the Research Laboratories of TheAluminum Company of America . . . 5224 . Object of Tests . . . . . . . . . . . 52

25. Specimens and Materials . . . . . . . . 5226 . Apparatus . . . . . . . . . . . . 5527 . Procedure . . . . . . . . . . . . 5528 . Results and Discussion . . . . . . . . 57

VIII. SUMMARY AND CONCLUSIONS . . . . . . . . 6929 . Summary of Analytical Results . . . . . . 6930. Comparison of Analytical With Experimental

Behavior of Aluminum Tubes . . . . . 7131. Conclusions . . . . . . . . . . . 72

BIBLIOGRAPHY . . . . 75


9/86

LIST OF FIGURESNO. PAGE

1. System of Coordinates . . . . . . .. . . . . .. . 102. Element of Shell in Equilibrium . .... . . . . . . . 123. Sections of Deflected Cylinders . . . . . . . . . . . . . 214. Collapse-Coefficients; Round Cylinders With Pressures on Sides Only,

Edges Simply Supported; A= 0.30 ... . . . . . . . 245. Curves for Determining a-b Relation .... . . . . . . 266. Graphs for Determining a/b Ratios, Fixed Edges . . . . . . . . 287. Collapse-Coefficients; Round Cylinder With Pressure on Sides Only,

Fixed Edges; u = 0.30 ... . . . . . . . . . . . 318. Collapse-Coefficients; Round Cylinder With Pressure on Sides and Ends,

Edges Simply Supported; p = 0.30 . . . . . . . .. . 329. Collapse-Coefficients; Round Cylinder With Pressure on Sides and Ends,

Fixed Edges; I = 0.30 . . . . .. . . . . . . . 3410 . Typical Stress-Tangent Modulus Curves .. . . . . . . 4211. Thin-walled Pipe for Collapse Test . ... . . . . . . 4512 . Section Through Iron-Pipe Bulkhead ... . . . . . . . 4613 . View of 20-inch Steel Pipe After Collapse . . . . . . .. . 4614 . View of Adjustable Bulkhead, Manometer, Air Ejection Pump,

20-inch Steel Pipe . . . . . .. . . . . . . . . 4715 . View of Collapsed 20-inch Steel Pipe and Calipers . . . . . . . . 4816 . Radial Deflection Curves-20-inch Steel Pipe With Pressure on Sides Only;

A. = 0.22 Inches . . . . . . . . . . . . . . .. . 4917 . Radial Deflection Curves-20-inch Steel Pipe With Pressure on Sides Only;

A. = 0.48 Inches . . . . . . . . . . . . . . .. . 5018 . Radial Deflection Curves-18-inch Steel Pipe With Pressure on Sides Only 5119 . View of 18-inch Steel Pipe After Collapse . . . . . . .. . 5220. Section Through Aluminum-Tube Bulkhead . . . . . . . . . 5321 . Arrangement of Test Equipment and Specimen for Tubes Requiring Pres-

sures Less Than 14 lb. per sq. in. to Produce Collapse . . . . . . 5422. Arrangement of Test Equipment and Specimen for Tubes Requiring Pres-

sures Greater Than 14 lb. per sq. in. to Produce Collapse . . . . . 5623. Method of Measuring Diameter, 6-inch O.D. Tubing . . . . . . . 5724. Specimens 2, 3, 4, and 5 After Test, and Low Pressure Apparatus. . . 6025. Specimens 6, 7, 8, and 9 After Test . . . . . . . . .. . 6126. Specimens 11, 12 , 13 , 14, 15, and 16 After Test . . . . . . . . . 6227. Specimens 20, 21 , and 22 After Test . . . . . . . . .. . 6328. Specimens 23, 24, 25, 26, 27, 28, and 29 After Test . . . . . . . . 6329. Radial Deflection and Maximum Stress Curves, Specimen No. 17,

Welded Aluminum Alloy Tube . . . . . . . . . . . . 64


10/86

6 LIST OF FIGURES (CONCLUDED)NO. PAGE30. Radial Deflection and Maximum Stress Curves, Specimen No. 18,

Welded Aluminum Alloy Tube . . . . . . . . . . . . 6531. Radial Deflection and Maximum-Stress Curves, Specimen No. 1,

Welded Aluminum Alloy Tube ... . . . . . . . . 6632. Radial Deflection and Maximum Stress Curves, Specimen No. 10a,

Extruded Aluminum Alloy Tube . . . . . . . . .. . 6733. Radial Deflection and Maximum Stress Curves, Specimen No. 10b,

Extruded Aluminum Alloy Tube . . . . . . . . .. . 68

LIST OF TABLESNO. PAGE

1. Tensile Properties of the Aluminum Alloys in the Tubes and Stiffeners . . 532. Collapsing Pressures of Thin-walled Aluminum Alloy Cylinders, Pressure

on Sides Only . . . . . . . . . . . . . . . . 583. Collapsing Pressures of Thin-walled Aluminum Alloy Cylinders, Pressure

on Sides and Ends . . . . . . . . . . . . .. . 59


11/86

A STUDY OF THE COLLAPSING PRESSURE OFTHIN-WALLED CYLINDERS

I. INTRODUCTION1. Resumg of Previous Work.-The problem of determining the

external pressure at which a thin-walled cylinder will collapse con-fronts the designers of boilers, penstocks, vacuum tanks, and similarunits of construction. In the design of hydro-electric or water sup-ply projects, problems of determining the collapsing pressure of thin-walled pipe and of evaluating the effect of stiffeners upon the strengthof the pipe are frequently encountered. Many industries using dis-tillation processes under partial vacuum are confronted with theproblem of designing tanks to withstand external pressure. Thedesign of submarines involves the same problem under complicatedconditions.

Many experiments have been made to determine the collapsingpressures of small pipe'* such as boiler tubes2 or similar tubes 3 andof heavy-walled lap-welded steel pipe4, but the conditions encoun-tered in large flumes, submarines, and tanks have not been studieduntil recently. Such tests have been made at the A. 0. Smith Cor-poration 5, and at the U. S. Experimental Model Basin 6. Tests alsohave been made to determine the buckling strength of thin cylin-ders subjected to axial loads. Such tests were made for the purposeof obtaining a guide in estimating the strength of airplane fuselages8', 9,standpipe shells 10 , and to substantiate a new theory for the bucklingof thin cylinders under axial compression and bending."

Theoretical analyses of the behavior of cylinders under externalpressures have also been made by a number of investigators. Bryan 12obtained (in 1888) the expression for the collapsing pressure of longthin tubes by means of the energy criterion for instability. Southwell13obtained (in 1913) an expression for the collapse of short tubes whichshowed that such tubes may buckle into more than two lobes. Un-fortunately, his expression contained an unknown parameter. Avalue for this parameter was determined (in 1914) by G. Cook 15 forthe case of hinged edges and lateral pressure only. R. von Mises de-rived (in 1914) an equation for the collapsing strength of short thintubes simply supported at the edges, and subjected to lateral pres-sure only e, which did not contain any undetermined constants.*Later (in 1929) he extended his work to include both lateral and end

*This and similar numbers refer to the bibliography at the end of the bulletin.


12/86

ILLINOIS ENGINEERING EXPERIMENT STATION

pressures"7. In 1920 Sanden and Gunther 18 used von Mises' formulasin studying the behavior of stiffened thin cylinders under uniformexternal pressure. In 1922 Westergaard 20 presented a general con-ception of the buckling of elastic structures which included generalequations for the gradual buckling of imperfect specimens or thosewith eccentric loading. In 1929 Tokugawa presented a paper 19giving a derivation of a formula for the collapsing pressure of cylin-ders which includes a "frame factor" applied to the length. Thisframe factor is determined experimentally. T'he paper also includeda study of the effect of stiffening rings upon the strength of the shell.

2. Purposeand Scope of Investigation.-The purposes of the studyherein presented are(a) to analyze the elastic behavior of thin circular cylindrical

shells subjected to uniform external pressure, and to determine thepressure at which such shells collapse for simply supported and forfixed edges; extensions in the analysis are made for plastic behaviorof the material, for "out-of-roundness" of the cylinder, and fo rstiffening effects of ring stiffeners;

(b) to study experimentally the behavior of thin-walled tubesunder uniform external pressures, for comparison with the results ofthe theoretical analysis.

3. Acknowledgment.-The investigation herein reported is a partof the thesis presented by the author in partial fulfillment of therequirements for the degree of Doctor of Philosophy in Engineeringat the University of Illinois in 1936.

The analytical work in the thesis and the experimental work onthe steel cylinders was done under the general guidance of H. M.WESTERGAARD who at that time was Professor of Theoretical andApplied Mechanics at the University of Illinois. The experimentalwork on aluminum alloy cylinders was carried out in the ResearchLaboratories of the Aluminum Company.

The author wishes to express his gratitude to Professor H. M.WESTERGAARD for his advice and guidance; to M. L. ENGER, deanof the College of Engineering, and to F. B. SEELY, head of the De-partment of Theoretical and Applied Mechanics, for their encourage-ment and assistance; to the Aluminum Company of America for the"use of its facilities; to Dr. F. C. FRARY, Director of Research andMr. R. L. TEMPLIN, Chief Engineer of Tests of the Aluminum Com-pany of America, for their cooperation; and to Mr. C. DUMONT,also of the Aluminum Company for his assistance with the tests.


13/86

COLLAPSING PRESSURE OF THIN-WALLED CYLINDERS

In editing the thesis for this bulletin of the Engineering Experi-ment Station, W. L. SCHWALBE, Associate Professor of Theoreticaland Applied Mechanics, extended the analysis of the problem toobtain the effect of certain terms that were neglected in the sim-plifying assumptions made in the thesis. The analysis herein pre-sented, therefore, contains some additions to that presented in thethesis, but the final results are substantially the same. The authorgreatly appreciates the careful analysis given the subject by Pro-fessor SCHWALBE.

II. THEORETICAL ANALYSIS4. Basic Arguments and Assumptions.-The general procedurefollowed in this investigation is to derive expressions for the col-

lapsing pressures of round cylindrical shells of elastic materials, andto treat deviations from these conditions as extensions of the firstderivation.

The argument used in determining the collapsing pressures is asfollows: Consider the cylinder deflected into some shape such thatthe differential equations of continuity and equilibrium combined,together with the boundary conditions, are satisfied. If the externalforces necessary to hold the shell in the deflected position are inde-pendent of the magnitude of the deflections as long as they are sosmall that they do not materially change the general shape of theshell, then the shell is in a state of neutral equilibrium. The lowestpressure at which neutral equilibrium may begin is the critical orcollapsing pressure of the cylinder. Below the critical pressure theequilibrium is stable, above the critical value the equilibrium isunstable.

The assumptions involved in setting up the general differentialequations are as follows:(1) The shell is a round cylinder before buckling.

(2) The shell is of uniform thickness throughout.(3) The material in the shell is homogeneous and isotropic, and

is elastic according to Hooke's law.(4) The thickness of the shell wall is small compared to the

diameter, so that the distribution of normal stress over the thicknessmay be assumed as linear.

(5) As a consequence of the preceding assumption, the radialstress, aZ, is negligible compared to the circumferential and longi-tudinal stresses, and the radial shearing detrusions are zero.

(6) Displacements are small compared to the thickness so that


14/86


FIG. 1. SYSTEM OF COORDINATES

certain small quantities may be neglected. Neglections are indicatedin the development of the analysis.

5. System of Coordinates.-The system of cylindrical coordinatesused to define and locate any particular element of shell is indicatedin Fig. 1. The whole system is based on a reference cylinder theradius of which is the average mean radius of the actual shell, andthe longitudinal axis of which coincides with that of the actual shell.The origin of coordinates is taken on the cylinder of reference, atthe point of maximum radial displacement of the deflected shell,and midway between the ends of the cylinder.

The y-axis is parallel to the axis of the shell, and lies on the cylin-der of reference, positive toward the reader.

The s-axis lies on the circumference of a right section of thecylinder of reference, positive in a clockwise direction. The coordi-nate is s= RO where R is the radius of the cylinder of reference,and 0 is the angle subtended by s in a right section of the cylinderof reference.

The z-axis coincides with the radius of the cylinder of reference,the radial displacement z being measured positive outward.

The meanings of all symbols used in the subsequent pages aregiven in the following section.

6. Notation.-In the discussion the following notation is used:A = constant (indeterminate in magnitude at collapse), repre-

sents the maximum value of deflectionA. = maximum initial departure from a round cylinder


15/86


a, b, c = arbitrary numerical constants determined from boundaryconditions and the general differential equation

B, C = constants determined by the differential equationsD = diameter of cylinder, inchesE = modulus of elasticity of the material in the shell or in the

stiffener, lb. per sq. in.E' = effective modulus of elasticity, lb. per sq. in.El = tangent modulus for the average stress, S 1, in the plasticrange, lb. per sq. in.EI, = flexural rigidity of combined stiffener and shell in length,

L,, lb.-in. 2t3I = - = moment of inertia per unit of length of shell, in.12

I, = moment of inertia of combined stiffener and plate, in.4K = numerical coefficient dependent upon the ratios of L/D and

D/t such that W, = KE (--DK 1, K 2, K', K", Ki', K 2 ', Ki", K 2 " = numerical coefficients, depend-ent upon the ratios of LID and D/t used in evaluating KL = length of cylinder, inchesLo = length of one wave under end load only, inches

L, = stiffener spacing, inchesM,e = bending moment in the stiffener, in.-lb.

N = number of lobes into which the shell collapsesP = end load in addition to external pressure, lb. per linear inchPc = end load causing collapse, end load only, lb. per linear inchQ = the elastic limit of the material in the shell, lb. per sq. in.;

this value may be taken as 1.1 times the proportional limitas determined by Tuckerman 32R. = radius from center of cylinder to the centroid of a right sec-tion of the stiffener and the plate effective with it, inches

r, r' = numerical ratios dependent on R/L and N introduced byloading conditions

S = maximum total stress in the shell (direct stress and bending),lb. per sq. in.

S = allowable total stress in an out-of-round shell, lb. per sq. in.S1 = average stress in the shell, lb. per sq. in.Sc = P,/t = unit longitudinal stress corresponding to Pc, lb. per

sq. in.Sp = unit longitudinal stress corresponding to an axial load

whether tension or compression, lb. per sq. in.


16/86

12 ILLINOIS ENGINEERING EXPERIMENT STATION

Notez - pos/itive downwvard:I -posit/ve toward reader.s - pos/'ive c/oc/w/kse.All// forces and coup/es -

pois//'-e as shown.

FIG. 2. ELEMENT OF SHELL IN EQUILIBRIUM


17/86


S. = modulus of failure of the material in the shell; the tensilestrength of the material may be used for S., lb. per sq. in.

S, = yield strength of the material, lb. per sq. in.S, = maximum circumferential bending stress in the deflected

shell, lb. per sq. in.t, = thickness of unstiffened shell having the same strength as

the stiffened shell, inchesW = the actual pressure applied, lb. per sq. in.W = allowable pressure of an out-of-round shell, lb. per sq. in.We = the collapsing pressure for the shell, lb. per sq. in.W, = external pressure at collapse of the stiffened shell, lb. per

sq. in.W. = ultimate collapsing pressure of an actual imperfect tube,

lb. per sq. in.Z, = radial deflection of the stiffener, inchesZo = initial departure from a round cylinder, inches(N 2L2a = 2--- + 1 = numerical ratio dependent on R/L and NOE = slope of the tangent to the stress-strain curve at the yield

strength, lb. per sq. in. (03= 0.10 to 0.20 for materials notcold worked, and 3 = 0.05 to 0.10 for materials cold workedappreciably)7,, = unit-detrusion in a tangential planee, = unit-strain in the circumferential directione, = unit-strain in the longitudinal direction

X =/ NL- 1 = numerical ratio dependent on R/L and N1 = Poisson's ratioe, = unit-strain in radial directionz = radial deflection of middle surfaceu = longitudinal displacement of a point in the middle surfacev = circumferential displacement of a point in the middle surface.

7. Deformed Element of Shell.-The element of shell considered isshown in its deformed state* in Fig. 2. If u, v, and z are the dis-placements in the directions y, s, and z, respectively, of a point inthe middle surface of the shell from the undeformed position, thenthe strains and the detrusion of the middle surface are defined as

au av v au OvS--= , f., = -- , "y, -- + . (1)

_y as R as Oy*A more complete description of a deformed element is given in Reference 34, p. 77.


18/86


By eliminating u and v, the following equation of compatibility 21, 22is obtained: 02e 02e 02a278 1 a2z+ - _ _ = 0. (2)ay 2 Os2 asay R ay2

56 is the infinitesimal angle at the center of curvature of thedeflected shell, subtended by the circumferential element of length.It may be expressed as a function of 0; , = dO, and consists00of the following parts:

(1) dO, original angle1 32z(2) - -- dO, due to change in slope over length dsR 002z(3) -- dO, due to radial deflectionR

(4) e,dO, due to circumferential strain.Hence I# 02z z-- = 1- ---- +e. (3)00 R 002 R

8. Internal Forces and Couples.-In Fig. 2 are shown the forcesand couples holding the element of shell in equilibrium.External Force:

W = external pressure, lb. per sq. in.Normal Forces:

(Positive when acting in the positive direction on the face of anelement facing the positive direction)P, = normal force acting on a unit of length of the face of the ele-

ment lying in a plane normal to the circumferential tangentto the shell, lb. per linear in.P, = normal force acting on a unit of length of the face of the ele-

ment lying in a plane normal to the longitudinal tangent tothe shell, lb. per linear in.


19/86


Shearing Forces:(Positive when acting in the positive direction on the face of an

element facing the positive direction)S,y = shearing force in the y direction on the unit of length of the

face of the element lying in a plane normal to the circum-ferential tangent to the shell, lb. per linear in.

Sy, = shearing force in the s direction on a unit of length of theface of the element lying in a plane normal to the longitudinaltangent to the shell, lb. per linear in.

S, = shearing force in the z direction on a unit of length of theface of the element lying in a plane normal to the circum-ferential tangent to the shell, lb. per linear in.

Sy, = shearing force in the z direction on a unit of length of theface of the element lying in a plane normal to the longitudinaltangent to the shell, lb. per linear in.

Couples:(Positive where the force of the couple farther away from the

center of the shell is positive)M, = bending couple or moment resulting from the distribution of

normal forces on a unit of length of the face of the elementlying in a plane normal to the circumferential tangent to theshell, in.-lb. per in.

M, = bending couple or moment resulting from the distribution ofnormal forces on a unit of length of the face of the elementlying in a plane normal to the longitudinal tangent to theshell, in.-lb. per in.

M,y = twisting couple or moment resulting from the distribution ofshearing forces on a unit of length of face of the elementlying in a plane normal to the circumferential tangent to theshell, in.-lb. per in.

M,. = twisting couple or moment resulting from the distribution ofshearing forces on a unit of length of the face of the elementlying in a plane normal to the longitudinal tangent to theshell, in.-lb. per in.

9. InternalFortesand Couples in Terms of Displacement.-On thebasis of Hooke's law the normal and shearing forces may be expressedin terms of the strains and detrusion as follows: 21 22, 34

EtP" = (=, +. ,)1 -


20/86


EtP. = - (e, + py,) (5)1 - p

2

EtSa = ) y,,. (6)The moments in terms of displacement, z, are:

El ( 2z p 02z z (My = -- - + - + - (7)1 - \ y2 R2 002 REl 1 a2z z 02zM. = - - (-- + - + -y (8 )1 - IA2 R 2 a62 R 2 ay2 )EI 1 a2zM., = -- - (1 - () )1 - A 2 R O9Qy

In the derivation of Equation (9) it is assumed that the dif-ference between M,y and My, is negligibly small. If this assumptionis not made, the left side of Equation (9) becomes Y (M, + My,).

The values for e,, ey, -y from Equations (4), (5), and (6) are sub-stituted into Equation (2). The result is

a2P, 02Pj 1 02Py 2aP,- + _ay 2 ay 2 R2 02 R2 90 2

2 (1 + I) a2Sa, Et 02z- (10)R M06y R (y10. Equationsof Equilibrium.-Theconditions of equilibrium ap-

plied to the force system shown in Fig. 2, neglecting differentials ofhigher order than the second, give the following equations:

OP. + Sy, a4- + + - = 0 (11)as ay OsaP.. OS... 02z-.. S, -- =0Oy as Oy2


21/86


OS., asy, 0a4' z az2z+ - W +P--P- -S, Sy (13)as ay as y2 1sayy)saMy 9M,.

SW= + -- (14)ay asaM, aMy,S az- (15)Os 9y

SM 8Mz-Sy -SyM + My. - = 0. (16)as ay 2

If products of forces and moments with the derivatives of dis-placements, which are small by comparison with unity, are neglected,and the shearing forces are eliminated from Equations (11) and (12)by means of Equations (14), (15), and (16), then, with M,y = My,,the following equation is obtained:

a2P, a2Py 1 + E, aM,. 2 (1 + e.) a2M.,+- +- = 0. (17)Os2 ay 2 R as2 R OsOy

Equations (14) and (15) are combined with Equation (13); thena2My a2M, 1 a,M. 9-- +2 + -= W+P,--ay 2 asay Os2 Os

a2z2z 02z-P y -- Sy - Say-- . (18)ay 2 OsOy OsOy11 . General Differential Equations.-The moments from Equa-

tions (7), (8), and (9 ) are substituted into Equation (18); thenEl 4z a2z 2 84z++R1 - Ap \ ay 4 R 2 y2 2 R a02 y2

1 a4z 1 02z 1 Oi9+ - + = W + P-R 4 a04 R 4 092 ) R 0692z 1 O2z 1 82z- Pay- - Sy. y Sy. R y (19)Oy2 R O99y R O00y


22/86


A second equation is obtained from Equations (17), (8), and (9):1 021P, 2P, (1 + e,) E l ( 1 04zR2 0'02 y2 R (1 - j2) R4 a64

1 02z (2 - M) )^z+- 4 02 R 2 0+2 Y = 0. (20)A third equation is obtained from the equation of compatibility (10),a2zcombined with Equation (12), in which the small quantity Sy,--is neglected:Oa , a2P, 1 O2P, y 02P. Et 02z-- + (2 + ) --- - - = - - (21)Oy2 ay 2 2 2 02 R Oy2

A fourth equation is obtained by differentiating the strain1 1 /9v-, (P -,P ) =- -+zEt R (0a /

twice with respect to y, giving02P , 02P, El ( 3v 2 (22)

y2 A+ 9 (22)9y y R Vy80 yThe simultaneous Equations (19), (20), (21), and (22) represent

the relations between the radial and circumferential deflections zand v, and the forces P,, Py, Sy., S,, and W, for a thin cylindricalshell which is not stressed beyond the elastic limit and which hassmall deflections.

To determine the external pressure W, the end load P,, theshearing stress S,y (or S,,), or any combination of these forces atwhich the equilibrium of the shell becomes indifferent, the shell isconsidered in the deflected shape which satisfies the foregoing dif-ferential equations and the boundary conditions. The function, z,representing this deflection must meet the requirement that itsmagnitude is indeterminate for some value of the external force.This value of the external force at which the condition of neutralequilibrium prevails is the critical load on the cylinder.

18


23/86


III. SOLUTION OF BUCKLING EQUATIONS FOR ROUNDCYLINDERS, WITHOUT STIFFENERS

12 . Uniform Pressure Applied to Sides Only.-For this type ofloading the value of the circumferential force per unit length ofshell, P8, may be expressed as

P. = - WR + f y, s).WR is the average value of P, and f is a function of y and s whichexpresses the variation of P, from the average value. When thedeflection, z, of the shell is very small, then f (y , s) is also very small.The longitudinal force, P,, has an average value of zero, for thisloading and the value of P, at any point departs from this averageby a small amount g (y, s), dependent upon the deflection z. Thisgives

P = 0 + g (y, s).The shearing forces S,, and S,. have average values of zero for

this case of loading with variations from the average value byamounts h (y, s) and j (y, s), respectively:

S, = 0 + h (y, ) S,, = 0 + J (y,s).The values of P,, Py, S,y, and Sy, are substituted into Equations

(19), (20), (21), and (22), and products such as f with terms in9^ S0z 02z z-- other than unity, g - , h --- , and j -- are neglected.90 9say s3y 9s3yThen

E l 4z a2-z 2 04z 1 ('z 04z++ R +1 - 2 Ly 4.2 R y, R2 Oay -- 002 0Z/J1 1 a 2z zR R - (23)

1 a 2f 02g (1 + ,) El (1 04 zR 2 2 y R3 (1 - 2) R2 004

1 02z a Z+ - -- + (2 - ) - = 0R2 a02t d'dy2 /


24/86


a2g 1 a'g a2f J a2f Et a2 z(2 + )-- +-- +- - -R =- (25)ay 2 R2 002 by2

R2 002 R By

2

02f 2ag Et ( v a0z ) _ = + (26)0y2 ay 2 R 0y20 0 by2 /(a) Edges of Shell at Ends Simply Supported.-The boundary

22z 02zconditions for z are z = 0, - = - = 0 for all values of 0 whenay 2 a62L fzy = -. Because of symmetry - = 0 for all values of 0 when2 z 00

y = 0, and -- = 0 for all values of y at 0 = 0.90 v(vFor the circumferential displacement, - = 0 for all values of 0L 00

at y = --These conditions suggest solutions of the form,

ryz = A cos NO cos --L(27)vyv = B sin NO cos --L

Figure 3 shows cross-sections of a shell deflected in various waysso that N = 2, 3, and 4, giving respectively two, three, and fourlobes. The number of lobes has been found to depend upon theproportions of the shell and for any given shell will be that numberwhich will give the lowest pressure at neutral equilibrium.

When z and v are substituted into Equations (23), (24), (25),and (26) the following equations result:

E l r 4 (2N2 - ) 7r N 2 N 4 -ry--- -- -- - A cos NO cos -1 - /A L2 RR R 4 R 4 L

W 2y 1= -- (AN 2 + BN) cos NO cos - - f (y , s) (28)R L R


25/86


Edrges Simp/y Supported' Edges Fie'dSymmetri/ca/ About < Symmetrica/ About &FIG. 3. SECTIONS OF DEFLECTED CYLINDERS

1 a2f 02g (1 - e,) EI [N 4 N2R2 002 Oy2 R3 (1 - 2) R . R2

+ (2 - ) - A cos N cos - (29)L 2 J La2g 1 a2g yf(2 + ) - + - - + -4y 2 R 2 002 Oy211 af Et r2 ry

- - = -- - A cos NO cos -- (30)R2 002 R L2 Layf 02g El 7r-- - = -- -- (BN + A) cos NO cos-. (31)By2 ay 2 R L2 L


26/86


From Equation (28) it follows that f (y, s) has the form

f (y, s) = C cos NO cos --Land from Equations (29) and (30) that g (y, s) has a similar form,

g (y, s) = D cos NO cos --LFrom Equation (31)

RBN + A =-- (C - pD).EtThe values of C and D are found to beC EtA Roa

Et WR\ a+l+u- [N'{1+(X-1) (2-u)-1 ] 1R3(1 -~) Et XaD EtA RXa

Et / WR\ l-M (a-l)+ [N2{1+(x-1) (2--)}-1] 1( E) - (-R3(1 -2) Et Xa

in whichN 2L 2

and N 2L 2a =--+ 1.72R2


27/86


From Equation (28)

Et - pEI- (-.l-) [N21+(X-1)(2-)} -1] A cos NO cos--R2a' R'(1 -_ ) ha LEl ry W ry+ EI N2 [N22-(X -1) -] A cosNOcos- =--FA cosNOcos- (32)R4 (1 -,2) L R L

where1 .

F=N-1 - + a 2 XaR-( 1 - [N2{1 +(X- 1)(2-A)} -1] 1-- [a(1-Ag)+ (1+A)2]+a+1+.R2(1-/.2)tXa1 E,)

Equation (32) indicates that solutions, different from zero, existonly if

Et El+-- N2-N2X -,(X-1)-1}Ra 2 R 3 (1 -. 2)

- +l [N {1+(X- 1)(2--.)}-1 ]W = W,= F . (33)

As the uniform external pressure on a round cylinder increasesfrom zero to the value W, the cylinder remains round and in stableequilibrium until W reaches the critical value We. At that pressuresmall variations in the internal forces f, g, h, and j become possiblewith indeterminate deflections of the cylinder from the round. Theboundary values for displacements at the ends come into effectanalytically only at the critical load. Actually, on account of thebulkheads (Figs. 12 and 20) the values of z and v at the ends areapproximately zero during the whole range of values of W fromzero to We.


28/86


Values of LFIG. 4. COLLAPSE-COEFFICIENTS; ROUND CYLINDERS WITH PRESSURES

ON SIDES ONLY, EDGES SIMPLY SUPPORTED; / = 030

t3Since I = -- an12

in which

D-= , Equation (33) may be written as2

t3 tWe = KiE- + K2 E--D3 D

a+- 1+--N2{N2W-,(X-1)- l- [N'{1+(X-1)(2-A) -1]2 aX

F(1 -M 2)

2K 2 = --afF

(34)

(35)

(36)


29/86


For the range of values D/t and L/R considered in this bulletin,F may be approximated by N 2 - 1. In order to plot values whichgive comparisons for various values of N, Equation (34) is written as

W. = (K +K 2 E-- = KE( . (37)t2 D)3

This form of the equation is convenient to use, whether the valuesof K be plotted in the form of charts, or arranged in tabular form.

Equation (37) gives the uniform external pressure at which around cylinder may collapse into N lobes. The number of lobesgiving the minimum value of W has been found by plotting curvesfor K and L/R for various values of D/t and N. Figure 4 shows afamily of K-curves for A = 0.30. The curves are shown only in theregion of the minimum value of K.

(b) Edges of Shell at Ends Fixed.-Boundary conditions for thisaz av Lcase are z = 0, - = 0, - = 0 for all values of 0 at y = +-ay 80 z 2

From symmetry and continuity, - = 0 for all values of 0 at y = 0Oz Oyand - = 0 for all values of y when 0 = 0. A solution of Equa-a0tions (23), (24), (25), and (26) satisfying these boundary conditions is

z = A cos NO cos -- + Ac cos NO cosh --aL aL (38)7ry Tryv = B sin NO cos -- + Bc sin NO cosh --bL bLFrom the boundary conditions and Equations (38) a relation betweena and b is found,

1 7r 1 r-tan- +-tanh -- = 0. (39)a 2a b 2b1The curves in Fig. 5a show the relation of - tan - to a and

1 i a 2a- tanh- to b. Values of a and b have been chosen from theseb 2bcurves so that Equation (39) is satisfied. The corresponding rela-tion between a and b is shown in Fig. 5b.


30/86


tql

Collapsing Pressure of Thin-walled Cylinders

Documents

Transcript of Collapsing Pressure of Thin-walled Cylinders